Thanks to all who responded. It seems that the most common notation for
the equivalence relation itself is E_0, from the literature on Borel
equivalence relations.
In addition to the responses on FOM, I received a few private email
messages. Kenny Easwaran pointed out that probabilists use the term "tail
set" for sets of equivalence classes under E_0. Franklin Vera Pacheco
suggested the term "almost equal" for the equivalence relation.
Tim